re-receiving old emails

With POP email, you have to delete emails on the server as they are not typically deleted. Often when you delete an email all you're doing is deleting it on the computer and not the server. POP is the original email handling protocol, and some email programs don't deal with deletions properly. IMAP is a more modern protocol and typically handles email better when you are using multiple devices to get your email.

I think the reason your provider is not seeing the problem is that their email deals with the email actually on the POP server and when they delete it, it's gone. You delete the email on your computer and it's still on the POP server. That's a tongue in cheek answer as different email programs deal with deleting differently. I've always used IMAP on my Mac's and don't know for sure how the Mac OS email program handles deletions. But this should give you a clue as to where to look. Hopefully :)

Have you previously deleted these emails? If not they only way they normally would show up is if you set up a new email application such that the computer redownloads the emails.

Another thing is if you are using POP or IMAP email protocol. Sometimes with emails you have to delete the email on every device. This has been pretty well fixed but it used to be a problem. You could delete the email on your phone and it would still show up on your laptop.

IMAP typically handles deletions better IMHO. Glad to be of assistance.